What programmable vector fields can (and cannot) do: force field algorithms for MEMS and vibratory plate parts feeders

Many man-made or natural objects are composed of symmetric parts and possess symmetric physical behavior. Although its shape can exactly follow a symmetry in the designing or modeling stage, its discretized mesh in the analysis stage may be asymmetric because generating a mesh exactly following the symmetry is usually costly. As a consequence, the expected symmetric physical behavior may not be faithfully reproduced due to the asymmetry of the mesh. To solve this problem, we propose to optimize the material parameters of the mesh for static and kinematic symmetry behavior. Specifically, under the situation of static equilibrium, Young’s modulus is properly scaled so that a symmetric force field leads to symmetric displacement. For kinematics, the mass is optimized to reproduce symmetric acceleration under a symmetric force field. To efficiently measure the deviation from symmetry, we formulate a linear operator whose kernel contains all the symmetric vector fields, which helps to characterize the asymmetry error via a simple ℓ2 norm. To make the resulting material suitable for the general situation, the symmetric training force fields are derived from modal analysis in the above kernel space. Results show that our optimized material significantly reduces the asymmetric error on an asymmetric mesh in both static and dynamic simulations.

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Emulation of Biological Motor Primitives in an Artificial System: The Generation of Static Force Fields

10.1115/imece1999-0122 ◽

1999 ◽

Author(s):

Chae J. Lee ◽

Bernard D. Reger ◽

Matthew C. Tresch ◽

J. Edward Colgate ◽

Ferdinando A. Mussa-Ivaldi

Keyword(s):

Spinal Cord ◽

Control System ◽

Force Field ◽

Vector Fields ◽

Force Fields ◽

Building Blocks ◽

Static Force ◽

Artificial System ◽

Leg Muscles ◽

Field Patterns

Abstract We have used observations of posture and movements in biological limbs to derive a controller for an artificial mechanism. The controller architecture emulates some of the known relations between spinal cord circuitry and the musculoskeletal system of vertebrates and, specifically, of the rat. This work relates to recent experiments suggesting that the neural circuitry of the spinal cord may be partitioned into a small set of functional modules. Activation of these modules, each connected to a set of limb muscles, resulted in force fields that have been measured at the endpoint of a limb. These force fields map each position of the foot into a corresponding static force vector. The force fields have been found to converge toward equilibrium positions located inside the leg’s workspace. The experimental observation that vector fields induced by multiple stimulations add vectorially, suggested that convergent force fields form a system of building blocks (or “primitives”) for the generation of stable postures and movements. To emulate this biological mechanism in the control of an artificial two-joint limb, we established relationships among three hierarchical levels — spinal modules, muscles, and actuators — by deriving the mappings among the respective output fields. These mappings are used in combination with an inverse model of the actuators to calculate the actuator commands that generate a desired force field. We tested the ability of this control system to reproduce the force fields generated by the leg muscles of the rat and a set of force fields with significant geometrical features. Our results show that we can successfully and reliably transfer to our artificial system the features of muscle force fields. In addition, we exploited the same principle of vector summation observed in the biological system to combine these muscle fields into a variety of force field patterns, including the gradients of Gaussian potentials and locally parallel fields. We consider this a first step in the generation of a biomorphic motor control system. This work is supported by ONR grant N00014-95-1-0571 and NIH grant MH48185.

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